Generally, to the occurrence of biochemical oscillation, we assume
that the system undergoes a supercritical Hopf bifurcation (HB) with
the change of a certain control parameter \(\mu \) . Eq.(REF )
has a unique stable point \(\mathbf {x}_{s}\)  with \(\mathbf {F}(\mathbf {x}_{s})\equiv 0\) ,
which loses stability at the HB point \(\mu =\mu _{c}\) , in the way that
the Jacobian matrix \(\mathbf {J}\)  with components \(J_{ij}=\left(\partial f_{i}/\partial x_{j}\right)|_{\mathbf {x}=\mathbf {x}_{s}}\) 
has a pair of conjugate eigenvalues \(\lambda _{\pm }=\alpha (\mu )\pm i\omega \) 
with \(\alpha (\mu _{c})=0\) . In the so-called supercritical region \(\mu >\mu _{c}\) 
(\(\alpha >0\) ), the deterministic system shows a stable oscillation
with frequency given by \(\omega \)  and amplitude growing from zero.
In the subcritical region with \(\text{$\mu <\mu _{c}$}\) 
(\(\alpha <0\) ), no deterministic oscillation can be observed. In the
case where the system size is not large such that the internal noise
term in Eq.(REF ) can not be ignored, such as for intracellular
biochemical oscillation systems considered here, an interesting phenomenon
known as noise induced oscillations (NIOs) has been observed even
in the subcritical region where \(\alpha <0\) , demonstrating the constructive
role of internal noise in mesoscopic chemical oscillation systems
[1]}. In addition, an optimal system size exists
where the NIO shows best performance, knows as internal noise coherence
resonance (INCR) [2]}, [3]}, [4]}.
